scholarly journals Lower Bounding the AND-OR Tree via Symmetrization

2021 ◽  
Vol 13 (1) ◽  
pp. 1-11
Author(s):  
William Kretschmer
Keyword(s):  
Lower Bound ◽  
Linear Program ◽  
Laurent Polynomials ◽  
Lower Bounding

We prove a simple, nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that ˜ deg(AND m ˆ OR n ) = ˜ Ω(√ mn ). We prove this lower bound via reduction to the OR function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [6, 10, 21]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson et al. [2].

scholarly journals A nonlinear Lazarev–Lieb theorem: L2-orthogonality via motion planning

2020 ◽  
pp. 1-17
Author(s):  
Florian Frick ◽  
Matt Superdock
Keyword(s):  
Lower Bound ◽  
Motion Planning ◽  
Unit Circle ◽  
Smooth Function ◽  
Unit Interval ◽  
Inner Product ◽  
Lower Bounding ◽  
Integrable Functions ◽  
Equivariant Topology ◽  
Norm Bound

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to [Formula: see text] can be simultaneously annihilated in the [Formula: see text] inner product by a smooth function to the unit circle. Here, we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain [Formula: see text]-norm bound. Our proof uses a variety of motion planning algorithms that instead of contractibility yield a lower bound for the [Formula: see text]-coindex of a space.

scholarly journals On the achievable rate of bandlimited continuous-time AWGN channels with 1-bit output quantization

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sandra Bender ◽  
Meik Dörpinghaus ◽  
Gerhard P. Fettweis
Keyword(s):  
Lower Bound ◽  
Mutual Information ◽  
Gaussian Noise ◽  
Continuous Time ◽  
Additive White Gaussian Noise ◽  
White Gaussian Noise ◽  
Information Rate ◽  
Achievable Rate ◽  
Zero Crossing ◽  
Lower Bounding

AbstractWe consider a real continuous-time bandlimited additive white Gaussian noise channel with 1-bit output quantization. On such a channel the information is carried by the temporal distances of the zero-crossings of the transmit signal. We derive an approximate lower bound on the capacity by lower-bounding the mutual information rate for input signals with exponentially distributed zero-crossing distances, sine-shaped transition waveform, and an average power constraint. The focus is on the behavior in the mid-to-high signal-to-noise ratio (SNR) regime above 10 dB. For hard bandlimited channels, the lower bound on the mutual information rate saturates with the SNR growing to infinity. For a given SNR the loss with respect to the unquantized additive white Gaussian noise channel solely depends on the ratio of channel bandwidth and the rate parameter of the exponential distribution. We complement those findings with an approximate upper bound on the mutual information rate for the specific signaling scheme. We show that both bounds are close in the SNR domain of approximately 10–20 dB.

scholarly journals A formulation of a matrix sparsity approach for the quantum ordered search algorithm

2017 ◽  
Vol 15 (02) ◽  
pp. 1750013
Author(s):  
Jupinder Parmar ◽  
Saarim Rahman ◽  
Jaskaran Thiara
Keyword(s):  
Lower Bound ◽  
Convex Optimization ◽  
Upper Bound ◽  
Optimal Algorithm ◽  
Search Algorithm ◽  
Quantum Algorithms ◽  
Semidefinite Program ◽  
Search Problem ◽  
Laurent Polynomials ◽  
Specific Subset

One specific subset of quantum algorithms is Grovers Ordered Search Problem (OSP), the quantum counterpart of the classical binary search algorithm, which utilizes oracle functions to produce a specified value within an ordered database. Classically, the optimal algorithm is known to have a [Formula: see text] complexity; however, Grovers algorithm has been found to have an optimal complexity between the lower bound of [Formula: see text] and the upper bound of [Formula: see text]. We sought to lower the known upper bound of the OSP. With Farhi et al. MITCTP 2815 (1999), arXiv:quant-ph/9901059], we see that the OSP can be resolved into a translational invariant algorithm to create quantum query algorithm restraints. With these restraints, one can find Laurent polynomials for various [Formula: see text] — queries — and [Formula: see text] — database sizes — thus finding larger recursive sets to solve the OSP and effectively reducing the upper bound. These polynomials are found to be convex functions, allowing one to make use of convex optimization to find an improvement on the known bounds. According to Childs et al. [Phys. Rev. A 75 (2007) 032335], semidefinite programming, a subset of convex optimization, can solve the particular problem represented by the constraints. We were able to implement a program abiding to their formulation of a semidefinite program (SDP), leading us to find that it takes an immense amount of storage and time to compute. To combat this setback, we then formulated an approach to improve results of the SDP using matrix sparsity. Through the development of this approach, along with an implementation of a rudimentary solver, we demonstrate how matrix sparsity reduces the amount of time and storage required to compute the SDP — overall ensuring further improvements will likely be made to reach the theorized lower bound.

Research on Exact Lower Bounding Segmentation of DTW Distance

2011 ◽  
Vol 255-260 ◽  
pp. 2267-2271
Author(s):  
Lin Feng ◽  
Chang You Xu ◽  
Hu Zheng ◽  
Bo Jin
Keyword(s):  
Lower Bound ◽  
Time Series Data ◽  
Distance Measure ◽  
Mining Area ◽  
Series Data ◽  
Time Warping ◽  
Related Field ◽  
Lower Bounding ◽  
Dynamic Time ◽  
High Computational Complexity

Dynamic time warping(DTW) distance is the most effective similarity measurement methods in time-series data mining area. Because of the high computational complexity of DTW, it is not suitable for huge amounts of data. Thus, some DTW distance lower bound methods are proposed, which can improve the efficiency of DTW distance calculation. In this paper, we summarized the existing lower bound of DTW method and then proposed a closely related field with the application and exacted Lower Bounding distance measure with Segmentation(eLBS). Experiment results show that this method meet the need of non-omission.

Lower Bounding Linear Program for the Perimeter Patrol Optimization Problem

2014 ◽  
Vol 37 (2) ◽  
pp. 558-565 ◽  
Author(s):  
Krishnamoorthy Kalyanam ◽  
Myoungkuk Park ◽  
Swaroop Darbha ◽  
David Casbeer ◽  
Phil Chandler ◽  
...  
Keyword(s):  
Optimization Problem ◽  
Linear Program ◽  
Lower Bounding

scholarly journals Incidences in Three Dimensions and Distinct Distances in the Plane

2011 ◽  
Vol 20 (4) ◽  
pp. 571-608 ◽  
Author(s):  
GYÖRGY ELEKES ◽  
MICHA SHARIR
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Three Dimensions ◽  
Sharp Bounds ◽  
Algebraic Analysis ◽  
Lower Bounding ◽  
Analysis Technique ◽  
Rigid Motions ◽  
Set Up

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new set-up, but are still unable to fully resolve them.Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes, Kaplan and Sharir [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in ℝ3. Applying these bounds, we obtain, among several other results, the upper bound O(s3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s3/k12/7).One of our unresolved conjectures is that this number is O(s3/k2), for k ≥ 2. If true, it would imply the lower bound Ω(s/logs) on the number of distinct distances in the plane.

A Fast Lower Bound for the Minimum Cost Perfect 2-Matching Linear Program

1993 ◽  
Vol 13 (3-4) ◽  
pp. 357-370
Author(s):  
Charles E. Noon ◽  
Guey-Mei You ◽  
Thomas J. Chan
Keyword(s):  
Lower Bound ◽  
Minimum Cost ◽  
Linear Program

A Lower Bounding Linear Programming approach to the Perimeter Patrol Stochastic Control Problem

2012 ◽  
Author(s):  
Krishnamoorthy Kalyanam ◽  
Swaroop Darbha ◽  
Myoungkuk Park ◽  
Meir Pachter ◽  
Phil Chandler ◽  
...  
Keyword(s):  
Linear Programming ◽  
Control Problem ◽  
Stochastic Control ◽  
Programming Approach ◽  
Linear Programming Approach ◽  
Lower Bounding ◽  
Stochastic Control Problem
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